Distinguishing Chromatic Number of Random Cayley graphs

The $\mathit{\text{Distinguishing}}\mathit{\text{Chromatic}}\mathit{\text{Number}}$ of a graph $G$, denoted ${\chi }_D(G)$, was first defined in \cite{collins} as the minimum number of colors needed to properly color $G$ such that no non-trivial automorphism $\varphi$ of the graph $G$ fixes each color class of $G$. In this paper, we consider random Cayley graphs $\mathrm{\Gamma }(A,S)$ defined over certain abelian groups $A$ and show that with probability at least $1-n^{-\mathrm{\Omega }(\log n)}$ we have, ${\chi }_D(\mathrm{\Gamma })\le \chi (\mathrm{\Gamma })+1$.